Step by step
The question
Question: find the area of the parallelogram with sides u = ⟨1, 1, 0⟩ and v = ⟨0, 2, 2⟩.
1 Set up the grid
Write a grid.
On top: i, j, k, for the x, y and z directions.
Under them, u, then v.
2 The x part
Cover the i column.
Four numbers are left: 1, 0, 2, 2.
Write them as a small grid.
Its cross: down 1 times 2, minus up 0 times 2.
2 minus 0 is 2.
3 The y part
Now cover the j column.
Four numbers are left: 1, 0, 0, 2.
Write them as a small grid, with a minus in front: the middle part always gets one.
Its cross: down 1 times 2, minus up 0 times 0.
Careful!
The y part, under j, always has a minus in front.
Forgetting it is the most common mistake.
2 minus 0 is 2.
With the minus in front: −2.
4 The z part
Now cover the k column.
Four numbers are left: 1, 1, 0, 2.
Write them as a small grid.
Its cross: down 1 times 2, minus up 1 times 0.
2 minus 0 is 2.
Put the three parts together: u × v = ⟨2, −2, 2⟩.
5 Find the length
The area of the parallelogram is the length of u × v.
Square the parts, add, take the root.
2 squared is 4.
(−2) squared is 4.
2 squared is 4.
Minus times minus is plus.
Add: 4 plus 4 plus 4 is 12.
But 12 has a square number in it: 4.
12 is 4 times 3.
A root of a times splits into two roots.
√4 is 2.
So ‖u × v‖ = 2√3.
That is the exact answer, made simple.
With a calculator, that is about 3.5, to the nearest tenth.
6 Check
Let's check the cross product: it is at a square corner to u and to v.
Both dot products are 0.
Right!
And the area is at most the two side lengths multiplied, about 4.
3.5 fits!
Let's look back.
First: set up the grid.
Then: the x part.
Then: the y part.
Then: the z part.
Then: find the length.
Last: check.